Average Modeling and Simulation of Series-ParallelResonant Converters by SPICE Compatible

Behavioral Dependent Sources

Sam Ben-Yaakov* and Gil Rahav

Tel: +972-7-461561; FAX: +972-7-472949; Email: sby@bguee.bgu.ac.ilDepartment of Electrical and Computer Engineering

Ben-Gurion University of the NegevP. O. Box 653, Beer-Sheva 84105, ISRAEL

Abstract - A new methodology for developingaverage models of resonant converters is presentedand verified against cycle by cycle simulation. Theproposed model ing approach applies the concept ofRac(t) which represent the instantaneous effective

load of the resonant network. Unlike the treatmentgiven in previous studies, the value of Rac isevaluated here dynamically as a function of thetemporal low-frequency-average of other relevantvariables . Once def ined, the model can be used as isto run steady state (DC), large signal (transient)and small signal (AC) simulations on any moderncircuit simulators. The proposed methodology wasused to develop a behavioral model of a series-parallel resonant converter. Excellent agreementwas found between simulation by the proposedmodel and cyc le by cyc le s imulat ion .

models derived by the proposed high level presentation, istheir ability to emulate the DC, large signal and small signalresponses of the corresponding switch mode or resonantsystem. Once derived, the models can be run as-is onpractically any modern circuit simulation package to obtainopen or closed loop responses in the time and/or frequencydomain. The fundamental ideas of the proposed approach areexemplified by developing the behavioral model of a series-parallel resonant converter and verifying the validity of themodel against cycle by cycle simulation. Following the sameline of reasoning, other resonant topologies can be capturedby analogous behavioral representation.

II. MODEL DERIVATION

Following Steigerwald [5], the basic operation of aresonant converter, such as a series-parallel converter (Fig. 1),can be represented by a damped resonant network (Fig. 2). Inthis representation the virtual AC resistor (Rac) expresses theeffect of the dissipative nature of the load (Rout, Figs. 1, 2)

on the resonant circuit. The value of (Rac), under steady state

conditions, can be obtained by equating the AC powerdissipated by it to the DC power delivered to the load (Rout).

This yields:

I. INTRODUCTION

Resonant converters have many favorable advantages. Theycan be designed for Zero Voltage Switching (ZVS), ZeroCurrent Switching (ZCS) in either current fed or voltage fedtopologies. Indeed, they were shown to be useful in amultitude of applications ranging from basic DC-DCconverters [1], active power factor correction circuits [2] tocapacitor chargers [3].

Rac= π2

8 Rout (1)A prerequisite for a solid engineering design of resonant

converters is a good model that describes their operation inthe time as well as in the frequency domain. Two basicapproaches have been used hitherto to develop such models.One approach applies analytical relationships to derive theexpressions that describe the behavior of a given converter inthe various domains [4]. A second approach developed bySteigerwald [5] uses the first harmonic approximation andthe Rac concept. By this, the converter is described as a

simple resonant network with a load dependent damping (orquality) factor which can then be examined by basic (steadystate) network equations. The limitation of the secondapproach is the difficultly of applying it to more than just thesteady state (DC) voltage ratio relationships.

where Rac and Rout are per the notations of Figs. 1, 2. It is

assumed that the switching frequency is above the resonantfrequency. For switching frequency below the resonantfrequency, the accuracy of the model may by poor [5].

Rac is in general time dependent. For example, in the

transient state, or in steady state when the switchingfrequency is modulated by a low frequency perturbation, the'load' seen by the resonant network is not constant. Underthese conditions, some reactive energy circulates in the outputfilter components. Yet, the average 'load' seen by the resonantnetwork at any given moment is resistive. This stems fromthe fact that the current through Lout (Figs. 1, 2) can be

considered constant over one switching cycle and the fact thatthe current drawn by the output section is always in phasewith the voltage across Cp (Figs. 1, 2) [5]. Consequently,

Rac can be considered as a time dependent resistor. The value

of Rac(t) at any given time can be derived dynamically by

In this study we overcome this deficiency of the Racapproach by extending the behavioral modeling methodology[6] to resonant converters. The advantage of the average

* Corresponding Author. Incumbent of the Luck-Hille chair ofInstrumentation design

2

dividing the average of the absolute value of the voltageacross Cp by the average current of Lout (Fig. 2). Namely: ERac =

π2

8 | |v(cp)-i(Ecp)

(Temporal value of Rac) (9)

Rac(t) = π2

8 | |Vcp(t)

ILout(t)

(2) Variable Reference

Figure

Coded Model Notation

(Fig. 3)where:

Rac(t) [Ω ] 2 Yes v(Rac) [Volt]

| |Vcp(t) [volt] 2,3 No v(cp) [Volt]

( | |Vcp(t) ) is the average of the absolute value of the voltage

across Cp and ( ILout(t) ) is the average current of Lout.

I Lout(t) [Amp] 2 No -i(Ecp) [Amp]

A basic assumption of the present modeling approach isthat quasi steady state conditions prevail during any givenswitching cycle. That is, we assume that rate of change of thedisturbances is sufficiently low such that steady statesolutions of the resonant network equations are a goodapproximation of the instant input to output relationships.Under this assumption, the average voltage across Cp can be

obtained by a simple steady state transfer function e.g.:

Vdc [Volt] 1 No v(in) [Volt]

f [Hz] Yes v(f) [Volt]

ω [1/Sec] Yes v(w) [Volt]

| |Vcp(t) = Vdc8

π2 | |H(jω ) (3) Table 1. Time dependent variables representation.

where:

| |H(jω ) = ωCsRac(t)

[ ]( )1-ω2LrCs2+( )ωRac(t)( )Cs+Cp-ω2CsCpLr

212

(4)

VdcCs

Cp

LrRoutCout Vout

Vm

Loutand Vdc is the DC input voltage.

Equations (2) and (3) can now be solved for | |Vcp(t) and

Rac(t) assuming that all other variables are known. In the

present approach, the chores of deriving the solution are leftto circuit simulators such as PSPICE [7] that have a build-incapabilities to handle behavioral dependent sources. Toaccomplish this, we first transform the problem into anequivalent circuit representation which is compatible withpractically all modern analog circuit simulators.In thisportrayal, all time dependent variables are coded into voltagesor currents (Table 1). Next, we present the relevant equationsby dependent sources that are a function of the coded variablesand constants. Finally, we add the excitation and the outputsection to complete the picture. The final result for the series-parallel converter is the equivalent circuit of Fig. 3. Thedefinitions of the independent and dependent sources are asfollows:

Fig. 1. Basis configuration of the series-parallel resonantconverter topology.

VcpRacCp

Lr Cs

Rout

Lout

Cout

IL

Ecp~ Vout

~Vin

out

Fig. 2. First-harmonic approximation of the series-parallelresonant converter.

It should be pointed out that the average model of Fig. 3 istransparent to the switching frequency. Namely, at steadystate, all the voltages and currents in the model (Fig. 3) areDC. During a transient state, the voltages and currents aretime dependent.

Vin = 4π Vdc (5)

Vf = f [Hz] (Switching frequency) (6)Ew = 2πv(f) [1/Sec] (Angular switching frequency) (7)

Ecp = v(in)(2/π)v(w)Csv(Rac)

[ ]( )1-(v(w))2LrCs2+( )v(w)v(Rac)( )Cs+Cp-(v(w))2CsCpLr

212

(Average voltage across Cp) (8)

3

For a constant switching frequency, the excitation Vf (Fig. 3)is a DC voltage source. For FM modulated switchingfrequency, Vf will comprise a DC component plus an ACcomponent that represents the frequency deviation. Intransient analysis Vf is time dependent.

A. DC Analysis

The 'DC' transfer function (output voltage as a function ofswitching frequency) was obtained by running a DC analysison the model. This was done by sweeping the voltage source(Vf, Fig. 3), which represents the switching frequency, overthe desired voltage-coded frequency range. Typical responsesfor different load resistors are depicted in Fig. 4. The originalresponse of the actual switching circuit was obtained byrunning a cycle by cycle PSPICE simulation, of the originalcircuit, in the transient (TRAN) mode. Each simulation runwas for one switching frequency and the asymptotic value ofsay, the output voltage, was used as the steady state solution.Comparisons between the results of the model simulation andthe cycle by cycle simulation (Fig. 5) reveal that the two arepractically identical. It is important to point out that the timerequired to obtain one point by the cycle by cycle simulationwas two orders of magnitude longer than the time required forthe complete DC sweep simulation by the model. Typicalrunning time (with a 33MHz 486 machine) were as follows.Simulation time for one point in cycle by cycle simulation :475 seconds; total running time for a complete DC sweep bymodel: 3.7 seconds.

1

V(f)

Vf 1

V(w)

Ew

V(in)

Vin

Rout

Lout

Cout

V(out)

-I(Ecp)Ecp

V(cp)

1 1 ERac

V(Rac)

Fig. 3. Average model of the series-parallel resonant converterby applying PSPICE (Micro Sim Inc) behavioraldependent sources.

0

105

210

315

80 110 140 170

Frequency [kHz]

Mod

420

200

Vou

t

Sim

Rout = 60Ω

Rout = 120Ω

III. RESULTS AND DISCUSSION

The proposed model methodology was verified bycomparing the model behavior against a full, cycle by cycle,PSPICE simulation.The parameters of the resonant converter studied were asfollows (Fig. 1):Vdc = 100V

Lr = 78µH Cs = 43nF Cp = 43nF

Lout = 1mH Cout = 1µF Rout = 15Ω − 120ΩThe comparison was made for steady state (DC), large

signal (transient) and small signal analyses (AC). Theprocedures and results for each case are given below:

Rout=15Ω

Rout=30Ω

Rout=60Ω

Rout=120Ω

100KV 150KV 200KV

400V

200V

0V

Frequency

Vou

t

Fig. 5. Comparisons between the DC gain results of modelsimulation (Mod) and the cycle by cycle simulation(Sim) for Rout = 120Ω and Rout = 60Ω.

B. Transient Analysis

The transient response obtained by the model was alsovalidated against a cycle by cycle simulation. We tested theeffect of a step in frequency. In the model, the frequency stepwas represented by a step in Vf. In the cycle by cyclesimulation, the switching frequency was instantaneousswitched from one frequency (155kHz) to another (165kHz).Typical results are given in Fig. 6.

Fig. 4. DC gain for different load resistors obtained byproposed model. Horizontal axis: 1V=1Hz

4

C. Small Signal Analysis 1. Frequency-perturbations to output-voltage.

An important advantage of the proposed model is theswiftness by which small signal analysis can be carried out.To do that, one defines Vf or Vin (Fig. 3) as a DC plus anAC voltage source and runs an AC analysis on the circuit.That is, the derivation of the linearized response is left to thesimulator. All that is needed is to properly scale the frequencyperturbations, i. e. 1V=1Hz. To verify the model behavior wecompare it also to cycle by cycle simulation. The latter isextremely tedious and time consuming. For each frequencypoint the switching frequency was modulated and after steadystate was reached, the modulation component at the outputwas examined for amplitude and phase.

The modulated signal (Vm, Fig. 1):

Vm = Vdc sgn[ ]sin 2πfst+βsin( )2πfmt (10)

where:Vref = cos( )2πfmt (11)

The amplitude and phase responses were then calculated by:

vac[dB] = 20Log

∆Vout(p-p)

2β fm (12)

ϕvac[deg] = ϕVout - ϕVref (13)

2. Line to output transfer function .

The modulated signal (Vm, Fig. 1):

Vm = [ ]Vdc+βsin( )2πfmt sgn sin( )2πfst (14)

where:Vref = sin( )2πfmt (15)

The amplitude and phase responses were then calculated by:

vac[dB] = 20Log

∆Vout(p-p)

2β (16)

ϕvac[deg] = ϕVout - ϕVref (17)M

agni

tude

[dB

]

-100

-2001.0KHz 100KHz 10MHz

fs=132kHz

fs=125kHz

fs=160kHz

Frequency

0

10Hz

.

10Hz 1.0KHz 100KHz 10MHz0d

180d

90d

fs=160kHz

fs=132kHz

fs=125kHz

Frequency

Phas

e [d

eg]

Fig. 6. Comparisons between Vout (Upper two traces) and Vcp(Lower two traces) response to a step in frequency.Upper traces: model simulation (Mod); Lower traces:cycle by cycle simulation (Sim).

Fig. 7. Small signal, switching frequency to output voltageresponse for different center frequencies as obtained bythe proposed model.

The relevant relationships for the cycle by cycle simulationare as follows [8]:

5

Results of typical simulation runs are given in Figs. 7, 8.The agreement between the model behavior and the cycle

by cycle simulation (Fig. 8) was found to be excellent.

IV. CONCLUSIONS

-30

-20

-10

0

10

1 10

Mag

nitu

de [

dB]

Mod

Sim

-180

Phas

e [d

eg]

1 10

Frequency [kHz]

-135

-90

-45

0

Phase

Magnitude

The behavioral modeling methodology for resonantconverters, developed in this study, was found to yield anaccurate model that checks well against cycle by cyclesimulations. The main advantages of the model are the ease ofits derivation and the fact that the basic average and high levelmodel is directly applicable to DC, transient and small signalanalysis. The derivation of the model is carried out for largesignal, leaving the task of linearization to the simulator.Following the same reasoning, similar models can bedeveloped for other resonant topologies.

REFERENCES

[1] R. L. Steigerwald, "High-Frequency ResonantTransistor DC-DC Converters," IEEE Transaction onIndustrial Electronics, Vol. IE-31, No. 2, pp. 182-90,May 1984.

-80

-70

-60

-50

1 10

-40 200

150

100

50

0

Phase

Magnitude

Phas

e [d

eg]

Mag

nitu

de [

dB]

Frequency [kHz]

Mod

Sim

[2] M. J. Schutten and R. L. Steigerwald, "Characteristicsof Load Resonant Converters Operated in High-PowerFactor Mode," IEEE Transaction on Power Electronics,Vol. 7, No. 2, pp. 304-14, April 1992.

[3] B. C. Pollard and R. M. Nelms, "Using The SeriesParallel Resonant Converter in Capacitor chargingapplications," APEC-92, pp. 731-37.

[4] V. Vorperian, "Approximate Small-Signal Analysis ofthe Series and the Parallel Resonant Converters," IEEETransaction on Power Electronics, Vol. 4, No. 1, pp.15-24, January 1989.

[5] R. L. Steigerwald, "A Comparison of Half-BridgeResonant Converter Topologies," IEEE Transaction onPower Electronics, Vol. 3, No. 2, pp. 174-82, April1988.

[6] S. Ben-Yaakov, "Average Simulation of PWMConverters By Direct Implementation of BehaviouralRelationships," Int. J. Electronics, Vol. 77, No. 5, pp.731-46, 1994.

Fig. 8. Comparisons between small signal responses. Uppertraces: switching frequency to output voltage response.Lower traces: Line to output voltage response (constantswitching frequency: fs = 160kHz). Model simulation

(Mod): circles. Cycle by cycle simulation (Sim):crosses.

[7] PSPICE: Micro Sim Co., Irvine, California.[8] J. Batarseh and K. Siri, "Generalized Approach to the

Small Signal Modeling of DC-to-DC ResonantConverters," IEEE Transaction on Aerospace andElectronic Systems, Vol. 29, No. 3, July 1993.